The relation between the reheating temperature, the number of e-folds and the spectral index is shown for the Starobinsky model and some of its descendants through a very detailed calculation of these three quantities. The conclusion is that for viable temperatures between 1 MeV and 109 GeV the corresponding values of the spectral index enter perfectly in its 2s C.L., which shows the viability of this kind of models ; This investigation has been supported by MINECO (Spain) Grant MTM2017-84214-C2-1-P and in part by the Catalan Government 2017-SGR-247. ; Peer Reviewed ; Postprint (published version)
We demonstrate that a wide range of viable ƒ(R) parametrizations (including the Hu and Sawicki and the Starobinsky models) can be expressed as perturbations deviating from the ΛCDM Lagrangian. We constrain the deviation parameter b using a combination of geometrical and dynamical observational probes. In particular, we perform a joint likelihood analysis of the recent type Ia supernova data, the cosmic microwave background shift parameters, the baryonic acoustic oscillations and the growth rate data provided by the various galaxy surveys. This analysis provides constraints for the following parameters: the matter density Ωm0, the deviation from ΛCDM parameter b and the growth index ɣ(z).We parametrize the growth index ɣ(z) in three manners (constant, Taylor expansion around z = 0, and Taylor expansion around the scale factor). We point out the numerical difficulty for solving the generalized ƒ(R) Friedmann equation at high redshifts due to the stiffness of the resulting ordinary differential equation. We resolve this problem by constructing an efficient analytical perturbative method in the deviation parameter b. We demonstrate that this method is highly accurate, by comparing the resulting analytical expressions for the Hubble parameter with the numerical solutions at low and intermediate redshifts. Surprisingly, despite its perturbative nature, the accuracy of the method persists even for values of b that are of O(1) ; S.N. acknowledges financial support from the Madrid Regional Government (CAM) under the program HEPHACOS S2009/ESP-1473-02, from MICINN under grant AYA2009-13936-C06-06 and Consolider-Ingenio 2010 PAU (CSD2007-00060), as well as from the European Union Marie Curie Initial Training Network UNILHC PITN-GA-2009-237920. This research has been co-financed by the European Union (European Social Fund—ESF) and Greek national funds through the Operational Program ''Education and Lifelong Learning'' of the National Strategic Reference Framework (NSRF)—Research Funding Program: THALIS. Investing in the society of knowledge through the European Social Fund. S. B. acknowledges support by the Research Center for Astronomy of the Academy of Athens in the context of the program ''Tracing the Cosmic Acceleration''
